Invariants
| Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $2 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B2 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}17&51\\8&7\end{bmatrix}$, $\begin{bmatrix}25&77\\68&59\end{bmatrix}$, $\begin{bmatrix}55&24\\4&77\end{bmatrix}$, $\begin{bmatrix}83&23\\32&73\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.36.2.s.1 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $32$ |
| Cyclic 84-torsion field degree: | $768$ |
| Full 84-torsion field degree: | $129024$ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.36.1-12.c.1.4 | $12$ | $2$ | $2$ | $1$ | $0$ |
| 84.36.1-12.c.1.3 | $84$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.